where
\(m_{L,R}=m\pm \frac{1}{2}\) .
To describe the spin 3/2 baryons (\(\Delta \)  resonances) in the soft-wall model at finite temperature
one has to introduce a pair of Rarita-Schwinger fields in the
bulk, \(\Psi _1^A\)  (for the left-handed spin-3/2) and \(\Psi _2^A\)  (for the right-handed spin-3/2), which obey the above
Rarita-Schwinger equations. The author has considered Ref [1]} to describe the spin 1/2 baryons (nucleons) profile functions in the soft-wall model at a finite temperature which obeys the Dirac equation in the
bulk [2]}. The left-handed spin-1/2 baryon profile function is given by \(F_1^A (r, T)\) , whenever the right-handed spin-1/2 baryon is given by \(F_2^A(r, T)\) . Note that the profile functions of spin 1/2 and 3/2 baryon are the same in a certain approximation.
The profile functions of \(\Delta \)  baryon \(f_{m}(r,T)\)  and nucleon \(F_{n}(r,T)\)  obey normalization conditions as
\(\int _{0}^{\infty }dr e^{-\frac{3}{2}A(r)}f_{m}^{L,R}(r,T)F_{n}^{L,R}(r,T)=\delta _{mn}\) 
